Basics: Calculus, Linear Algebra, and Proof Writing

This is the first section of the list of best math books. The “Basics” includes the best calculus book and best linear algebra book lists. There are also lists of some good books on proof writing and how to study mathematics for math majors, as well a collection of great math books that don’t require a lot of prerequisites. If you want to become a (pure) mathematician, this is the place to start.

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Best Calculus Book List

All of the books on this list are great, and if you read anyone of them you’ll really understand calculus. Most people will get more benefit from these as a second course in calculus than as a first exposure to the subject. Of course, some students could jump straight to these with no problem. It’s up to you.

Spivak, Calculus.
This is a great book for single-variable calculus. If you are an enthusiastic high school student or an honors track freshman math major, this is the book I would absolutely recommend. More than the others on this list, this book is fun, inspiring, and gives a sense of how to enjoy doing mathematics. I think the other calculus books are great (and a better choice for other types of students), but if you’re just starting out in abstract, rigorous proof-based mathematics, this is the way to go. It will clear up the misconceptions you learned in a first calculus class, and prepare you for undergraduate real analysis. The two major drawbacks to Spivak’s text are that it only covers single variable calculus so you’ll have to find another book for multi-variable. (Spivak did write a follow up, Analysis on Manifolds, but it never felt quite like a true multi-variable calculus book to me, and I didn’t love it overall). The other downside is that there are no applications of calculus at all. One last upside to this book is that if you’re looking for the best calculus textbook for self study this is a great choice since Spivak authored a solutions manual to accompany this edition and the previous edition. You can buy a copy of Spivak here.

ApostolCalculus – Volume 1andCalculus, Volume 2.
As readers of the blog know, I’ve read these two volumes cover to cover and have done every exercise. While Spivak is the calculus book I would recommend for high school students and beginning college students, Apostol is the text I would recommend for students with a bit more mathematics experience, or older students coming back to mathematics. It’s not as fun and inspiring, but is comprehensive, rigorous, and complete. This is the best calculus book for building mathematical maturity. It is like advanced textbooks in its presentation: definition, theorem, proof. Learning to read this style of text is necessary for every math student since virtually all upper level texts will look like this.

Many people find this book overly dry. Apostol also has a tendency to give difficult and tedious calculation exercises in sections where he cannot come up with any conceptually difficult exercises. Of course, doing a couple hundred of these will help you learn all of the common trig identities and stop making stupid algebra mistakes. (Nothing is quite so frustrating as spending an hour trying to figure out your mistake only to find out it was an basic algebra mistake.) The other oddity here is that Chapter 15 and 16 of Volume 1 are identical to Chapters 1 and 2 of Volume 2. So if you only want to buy Volume 1 then you’ll get a bonus 2/5ths of an introduction linear algebra. (I’ve never been able to understand the logic of this.) Despite these drawbacks, it is a fantastic book. If you read it cover to cover (including the introduction to Volume 1!) you will be way ahead for both your future real analysis and linear algebra class. Although Apostol did not write a solutions manual, if you are self-studying you can always look at my solutions to all the exercises in Apostol. (Sorry, shameless plug for the blog!) My solutions are not as good as the ones Apostol would have written, but at least you can leave a comment and I’ll respond. These are great books. You can buy Volume 1 here and Volume 2 here.

This is without a doubt the best calculus book for engineers, physicists, or people looking for applications of calculus. Physics, engineering, or applied math students who want a rigorous calculus textbook with a lot of exercises showing how to apply calculus then this is the best calculus book for you.

If you’re self studying, there’s a small catch. Originally, all of the solutions to all of the exercises in this book were provided at the back of Volume 2. However, in the Springer “Classics in Mathematics” edition (which is the only one currently available that I know of) the solutions to Volume 1 are no longer included. So, the second half of Volume 2, Book 2 contains the solutions only for Volume 2. I’ve heard if you look for used copies of earlier editions from the 1960’s and 1970’s it’s possible to find the solutions for Volume 1, but I don’t know how easy that is to find. (Update: As pointed out in the comments on the main best math books page an old solution set from Arthur Blank is available for free online through archive.org. You can find it here.)

In addition to the problems with the solutions, this book has a confusing publication history that I’ll try to clear up. The three books listed above comprise Volumes 1 and 2 of Introduction to Calculus and Analysis (Volume 2 is split into two separate books). There is an earlier, similar two volume textbook Differential and Integral Calculus, Volume I & II, with only Courant as author, which was written by Courant in German in the 1930’s. These two sets are technically different books, but are very similar. That said, the version you want is the yellow one from Springer not the black set from Wiley. They are really almost the same, except the yellow ones move a little further down the road of elementary real analysis (and is cheaper for all three books versus the two volume Wiley classic edition of earlier textbook).

Hardy, A Course of Pure Mathematics.
First published in 1908 this is a classic by one of the masters of mathematical exposition. I tried to read it in high school and failed. The book is absolutely rigorous, but I think hard to read these days. Worth taking a look at. It was the Calculus book for several generations of mathematicians. If you manage to read the whole thing, you’ll have my admiration. There’s this gem from Hardy in the preface, which should give you a clue as to the intended audience, “It is in any case a book for mathematicians: I have nowhere made any attempt to meet the needs of students of engineering or indeed any class of students whose interests are not primarily mathematical.”

It is in the public domain so you can get a free pdf on Project Gutenberg here. There’s also a nice printed edition from Cambridge University Press available here.

Best Linear Algebra Book List

Linear algebra is incredibly important in both applications and pure math. If you stay in math you’re going to learn and relearn linear algebra several times. (More advanced linear algebra books will show up in the Core Subjects section.) These are the best linear algebra books for a first encounter.

Hoffman & Kunze, Linear Algebra.
Without any doubt at all this is my favorite linear algebra book. Also, my favorite book in the entire “Basics” section. This book is really fantastic, and made a big impression on me. I credit this book more than any other with helping me develop some mathematical maturity. Maybe I read it at just the right time (I was about finishing Volume 1 of Apostol, but hadn’t read much else). It does way more than any of the other introductory linear algebra textbooks I’m familiar with. Each chapter starts out gently and steadily increases in difficulty level. The last section of some chapters can become particularly challenging. (I had to skip the last half of Chapter 5 on the Grassmann Ring… Maybe I’ll get back to it one day.) I also found the exercises in this book perfectly done. Each section contains a few straightforward computations and easy proofs, and then another seven or eight challenging exercises that force you to really understand the linear algebra you’ve just learned.

If you want a linear algebra book for self-study, this one doesn’t have solutions. I plan on adding them to the blog, but that won’t happen until this coming summer (2016). (Yes, this is another textbook I’ve read cover to cover and for which I’ve done all the exercises.) You can get Hoffman & Kunze here.

Axler, Linear Algebra Done Right.
This book differs from other linear algebra books in that it avoids determinants (almost) completely. There is a certain logic to this: it focuses the text on the structure of linear operators and finite dimensional vector spaces. It is clear and readable, and many people really love it. I sort of like determinants, but this is still one of the best linear algebra books at this level. If you are looking for a linear algebra for self study this book is widely used, so you can find university course pages that will likely have a syllabus you can follow. The latest is the 3rd edition which you can get here.

Halmos, Finite Dimensional Vector Spaces.
Skinny little Springer UTM (Undergraduate Texts in Mathematics), first published in 1958, this is an extremely elegant presentation. Of course, Halmos is one of the greater writers of mathematics and it certainly shows here. However, this book does not cover the material of a standard linear algebra class. Probably best as a supplement to one of the other books on this list. (Or if you want to go into functional analysis, you should certainly read it.)

Friedberg, Insel, and Spence, Linear Algebra.
This seems to be the standard choice for honors undergraduate courses in the US these days. It is more challenging than the usual computational type introductions to linear algebra. If you want something more applied and less theoretical than the above three books, this is the best linear algebra textbook for you.

Books on Mathematical Proof Writing

Writing mathematical proofs is critical for every math major. Upper-division math is almost entirely about writing proofs. The Calculus and Linear algebra books above are all proof-based, and hopefully you’ve had some exposure to proof writing before starting. If not, the books below would make very nice supplements. These are also great for anyone struggling in an analysis or abstract algebra class with how to write math proofs. I was a teaching assistant for an “Introduction to Proofs” type class at my university, and we used Velleman’s book. Most students really liked it, and I think it did a great job of helping students figure out proof writing.

Velleman, How to Prove It.
This is the best choice to learn to write math proofs. It’s great for everyone, but since Velleman comes from a Computer Science background it might be especially useful for learning proof-writing for computer science students. It starts out with sets and logic and the basics of proof techniques. It also contains complete solutions to all of the exercises, so is good for self-study. Either this or Hammack’s book would be a good supplement for getting started with proofs, or for discrete math classes.

Hammock, The Book of Proof.
A really good book in this genre. If you’re having trouble transitioning to proof-based math courses, or are self-studying math and realize you need more guidance on writing proofs, this is a good book. The book starts out carefully defining notation, sets, and logic and builds up to proof techniques and proof writing gradually. There are plenty of exercises in the text, with solutions to odd numbered exercises in the back of the book. This is a really good choice for aspirint math majors or for self-study.

Aigner & Ziegler, Proofs from the Book.
If you are learning to write proofs, and trying to understand what mathematicians mean by the word “elegant,” then examples of some of the most beautiful mathematical proofs throughout history is enlightening. The book’s title is a reference to “The Book” which Paul Erdos said contained only the most elegant proof of each mathematical theorem. Proofs from the Book presents 44 theorems, many with multiple proofs that are some of the most surprising, and elegant in mathematics. Go here if you want to be inspired by how beautiful a mathematical proof can be. You can get the fifth edition here.

Alcock, How to Study as a Mathematics Major.
Not strictly a book on proof writing, but this seemed like the most logical section to put this in. Alcock’s book is intended to help students with the mental transition from computational based math classes to abstract classes in analysis and algebra. I wanted to include it, because it is a really nice book, and something I wish I’d had when I was starting to study abstract math. It is well-researched and provides excellent insights in the way you need to think about abstract math, and how it differs from lower division math classes. I highly recommend this for math majors just starting with upper division or proof-based math classes.

5 comments

Hello! I am an undergraduate student but I want to go deeper in my Mathematical understanding and with a rigorous and proof-based systematic study. I was looking online for some suggestions on Calculus, Linear Algebra and Abstract Algebra books, but many different opinions in sites like Quora only made things more confuse. I really loved your site and found the suggestions more than helpful. I have other doubts regarding some books I was lookking for:

1) I own Piskunov’s Differential and Integral Calculus, but I want to rebuild my Calculus knowledge from base (I haven’t seen this subject since 2014), so I am inclined to go for Spivak first, then Apostol/Courant and then jump into other titles. Where does Piskunov’s book fit in?I hear that it is a more advanced textbook which focus on problem solving/applications rather than theory.